Black holesin

Gauss-Bonnet gravity

Dmitry G. Orlov (National Central University, Taiwan)

19 January 2009

in collaboration with C.M. Chen (NCU,Taiwan)

and D.V. Galtsov (MSU, Russia)APCTP-NCTSInternational School/Workshopon Gravitation&Cosmology

• Motivation

• Lovelock gravity

• Dilatonic black holes

• Dilaton Gauss-Bonnet gravity

• Summary and recent result

We consider an extremal limit of dilatonic black hole in Einstein-Maxwell-dilaton system. It’s know that such limit is bad defined: the event horizon become singular and shrink to single point,so entropy for such solution is also zero.

From other hand quantum calculation gives non-zero value of entropy.So it claims that corrections from more general theory may resolve this problem.

One of such correction is Gauss-Bonnet term:

which arise in two theories:• one loop correction in heterotic string theory• Lovelock theory

Motivation

2 4R R R R Rμν αβμνμν αβμν− +

Lovelock theory

D. Lovelock JoMP v.12, n.3, 498 (1971)

( ) 0 0ij ji ij ji ij ijij ab ab c ab cd j invacuoG A A A A g g g A A, , ;→ = , = , , , = , | =

LL action[ 2]

( )

0

dp

G pp

I Lκ α/

=

= ∫ ∑2 1 2 2 11 2

1

( ) ... ...p p p d

d

a a a aa apa aL R R e eε − += .L

But equation of motion possess in general, several constant curvature solutions with different radii, making the value of the cosmological constant ambiguous. In fact, the cosmological constant could change in different regions of a spatial section, or it could jump arbitrarily as the system evolves in time.

Some of solutions are ``spurious'' in the sense that perturbations around them yield ghosts.

Cristomo, Troncoso, Zanelli, PRD62 (2000) 84013, hep-th/0003271

2( )

( 2 )0

p k

kp p

kl p kc pd p

p kα

−⎧ ⎛ ⎞, ≤⎪ ⎜ ⎟:= = −⎨ ⎝ ⎠

⎪ , >⎩

12{1 [ ]}dk −∈ ,...,

22

1 ( 1)( 2)2( 2) 2d k

d dd G l

κ−

− −= ,Λ = −

− !Ω

These problems are overcome demanding the theory to have a unique cosmological constant.

The simplest class of such theory describe by action

( )

0

kp

G pp

I Lκ α=

= ∫∑ where

For a given dimension d, the coefficients give rise to a family of inequivalent theories, labeled by the integer k, which represents the highest power of curvature in the Lagrangian.

kpc

2

14

dM

d

I g F F d xμνμνε −

= − −Ω

∫

Einstein-Hilbert action

Maxwell term

( )

0

kk p

k pp

I c Lκ=

= ∫∑

LL term

2

1 ( 2 )2( 2)

dEH

d

I d x g Rd G−

= − − − Λ− Ω

∫

Simplest test for Lovelock theory is existing of exact solution

For system consists of

0 3( )( 3) d

QA rd rεφ∞ −= +−

22

2( ) 1 ( )krf r g rl

σ= + − ,( 1)( 1) kσ += ± ( )

12

2 12 1 2( 2)

2( )

3

k

k d k kk d k d k

G M G Qg rr d r

δ ε− ,− − − −

⎛ ⎞+= −⎜ ⎟⎜ ⎟−⎝ ⎠

22 2 2 2 2

22( )( ) d

drds f r dt r df r −= − + + Ω .

0 0 ( )r rF A r= −∂ .

Spherical symmetric black hole solution

electromagnetic field strength

, where

Metric:

Dilatonic black holes

System consists of gravity, form field and scalar field:

4 2[2]

1 1 e2 2

aS d x g R Fq

μ φμφ φ

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

= − − ∂ ∂ −!

∫

2 2 2 2 2 2 22e e eB A Cds dt dr d= − + + ΩFor the spherical symetric

case metric is:

e ag g F Fφμν μν φ φ−→ , → ∗ , → −

where ∗ denotes a d -dimensional Hodge dual,

so we will restrict ourselves to purely magnetic solutions (for the case one charge).

S-duality

[2] 2vol( )F b= Ω

• case of one charge (magnetically charged brane)

• dyonic case

[2] 1 2 2 2 2 2vol e vol vol vol( )aF b b φ−= + ∗ , = Ω

(analytic dyon solution known only for the cases of dilaton coupling )2 1,3a =

Black hole is supported by the form field

For the case of outer spherical symmetric space exist two type of BH solutions:• asymptotically flat• linear dilaton background (LDB) solution

Gal'tsov, Klevtsov, DO and Clement, IJMP A21,3575(2006) ; hep-th/0508070

Clement, Gal'tsov, Leygnac and DO, PRD73, 045018 (2006); hep-th/0512013

• asymptotycally flat solution - dilatonic black hole

2 2 21 1 22 2 21 1 1

220

02

2 2 1 2 2 22

[2] 22

e e

2 e vol( )1

a a aa a a

a

a

aa

ds f f dt f f dr r f d

f

r rFa

φφ

φ

− −+ + +

Δ

−−+ − + − −

−

−+ −

= − + + Ω

=

= Ω+

where

( ) 1 rfr

ξ ±± = −

( )2221

224

22

2

1

aa

a fR r f rfa

+−− +− −

−

⎡ ⎤= | | ⎢ ⎥

⎣ ⎦+

Scalar

curvature:

2222 21 1

24 ( )a

a aS r r rπ + ++ + −= Ω | | − - entropy

2221

aaR f +

−

−

But if we want to get extremal solution

r r− +→

We cann’t do it with getting regular horizon, because the resulting point , as we can see fromscalar curvature

r r r− += =

will be singular point and radius of horizon for such solution will be zero, just as entorpy.

By another hand quantum calculation give non-zero value of entropy.

• black hole on linear dilaton background (LDB)

2 22 21 1

2221

0

0

12 2 2 2 20

20

0

20

[2] 22

1 1

e e

2e vol( )1

a a

aa

aa

a

rr c cds dt dr r dr r r r

rr

rFa

φφ

φ

+ +

+

−⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

− /

⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞= − − + − + Ω⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠

⎛ ⎞= ⎜ ⎟

⎝ ⎠

= Ω+

221 2

22 2 2

0

2(1 )

ar a cR aa r r r

+⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟+ ⎝ ⎠⎝ ⎠

Scalar curvature:

(neutral solution on charged background)

12 2 2 2 2

1 21 1

1

[2] 1 1 2 2

1

2 ( )( )vol( )

a

f f f fds dt dr r f df f

ef

drF r r r r r r dtr

φ

−

+ − + −

+ − + −

⎡ ⎤= − + + Ω⎢ ⎥

⎣ ⎦

=

⎡ ⎤= − − Ω − ∧⎢ ⎥⎣ ⎦

• asymptotycally flat solution - dyonic black hole ( )2 1a =

Another solution for a dilaton coupling is received by Toda lattice method.

Both solution have a regular extremal limit, but if we consider a expansion on horizon, then we see that extremal limit exist only for some particular value of

a dilaton coupling

2 3a =

2 ( 1)1,3,6,11,...,2

n na +=

• dyonic solution - black hole on LDB2

2 2 2 20

0

0

[ ] 0 2 22 vol( )

k

a

n

rr drds f f dt r dr r f f

rer

drF r r rr

φ

⎧ ⎫⎪ ⎪⎪ ⎪⎨ ⎬+ − ⎪ ⎪⎪ ⎪+ −⎩ ⎭

+ −

= − + + Ω

=

⎡ ⎤= Ω −⎢ ⎥⎣ ⎦

2( 1)a =

• dyonic solution - black hole on LDB ( 0)a =2 2 2

2 2 2 22

0 0

[2] 0 20

1

2 vol( )

a

r r drds f dt r dr r f

e

drF r dtr

φ

−

++

⎡ ⎤ ⎡ ⎤ ⎛ ⎞= − + + Ω⎜ ⎟⎢ ⎥ ⎢ ⎥

⎝ ⎠⎣ ⎦ ⎣ ⎦=

⎡ ⎤= Ω − ∧⎢ ⎥

⎣ ⎦

(charged solution on charged background)

(neutral solution on dyon background)

DBH w GB term

4D theory. The action consists of gravity, scalar filed, Maxwell field and GB term:

2 2 4GB

1 2 e16

aS R F L g d xμ φμφ φ α

π⎧ ⎫⎪ ⎪⎛ ⎞⎨ ⎬⎜ ⎟⎪ ⎪⎝ ⎠⎩ ⎭

= − ∂ ∂ − − −∫

where Gauss-Bonnet term

2GB 4L R R R R Rμν αβμν

μν αβμν= − +

Chen, Gal'tsov, DO, PRD 75, 084030 (2007), hep-th/0701004Chen, Gal'tsov, DO, PRD 78, 104013 (2008), 0809.1720

Non-extremal solutionexterior solutionP. Kanti, K. Tamvakis Coloured black holes in higher Curvature string gravity, hep-th/9609003

T. Torii, H.Yajima, K. MaedaDilatonic black holes with Gauss-Bonnet Term, gr-qc/9606034

P.Kanti, N.E. Mavromatos, J. Rizos, K. Tamvakis, E. WinstanleyDilatonic black holes in higher curvature string gravity, hep-th/9511071

inner solutionS.O. Alexeyev, M.V. PomazanovInternal solutions in higher order curvature gravity, gr-qc/9706066

M.V. PomazanovOn the structure of some typical singularities for implicit ordinary differential equations, math-ph/0007008

Solution with a scalar singularity (with turning point

by Alexeyev, Pomazanov)

( )( ) ( )ss csR r O R r=

We consider static spherical symmetric solutions

parameterizing the metric as

22 2 2 2

2( ) ( )( )

drds W r dt r dw r

ρ= − + + Ωwhere

2( ) ( ) ( )W r r w rσ=

There are two common cases of gauge:1) 12) 1

σρ=′ =

For our purpose we choose first one.

The corresponding ansatz for the Maxwell one-form is

Like previously we consider two cases of system - with one charge and dyonic.

( ) cosmA f r dt q dθ ϕ= − − ,

Extremal case

Eqution of motion have strongly non-linear view, so we have only on way for investigation our model:

• analytic expansion on extremal horizon(amount of free parameters)

• getting global structure of solution by numeric integrating to infinity(we are looking for asymptotically good defined solutions)

• behaviour of solutions

•Analytic expansions

For general expansion the electric and magnetic charges are related as:

so it’s possible exit an electrically charged and dyonic solution, but not magnetically charged.

2 20

2

42(2 )

me

m

qq

qα ρα+

=+

2 2 4 2 2 2 2 3 42

2 4 2 2 2 2 22

22

( ) 3( 1) 6 (3 2) 4 (5 3) ( ) 76

( ) 1 ( 1) 2 (3 2) 4 ( 1) ( )4

( ) ( ),2(2 )

m m

m m

a

m

w r x a q a q a x O x

r a q a q a x O x

P r e x O xq

ϕ

ς α αα

ςρ α αα

αα α ςα

⎡ ⎤= − − + − + − + ,⎣ ⎦

⎡ ⎤= + − + − + − + ,⎣ ⎦

= = + ++

Near horizon expansion of solution:

where - define good behaviour for derivative of metric function on horizon.

1

1

PP

ς =| |

( )rρ

The expansion on space infinity:

2 13

2

2 223

2

23

2

2( ) 1 ( )

(2 ( ))( ) ( )2 3

2( ) ( )2

e m

e m

e

Q QMw r O rr r

D MD a Q QDr r O rr r

DM aQDr O rr r

α α

α αρ

αφ φ

−−

−

−∞

+= − + + ,

− −= − − + ,

−= + + + .

where the asymptotic value of the dilaton function.

, ,a ae e m mQ q e Q q eφ φ∞ ∞−= =

φ∞ −

• Numerical analysis for solution with null magnetic charge

Critical values of dilaton coupling constant:

0 488219703cra| | .

Two type of solution on space infinity:

• asymptotically flat solution

• solution with turning point

cra a| |<| |

cra a| |>| |

Asymptotically flat solution cra a| |<| |

The functions ( ) ( ) ( ) Hw x x P x x r rρ′, , , = − , for 0 1ρ = and some values of the dilaton coupling

constant: 0 1a = . ( 1 0 01P = − . ) — thin line, 0 4a = . ( 1 0 446P = − . ) — normal line, 0 45a = .( 1 1 18P = − . ) — thick line.

Solution with turning point cra a| |>| |

The metric functions ( ) ( ) ( )Hx w x r rρ ρ, , = − for cra a> . Numerical curves are presented for

0 5a = . and 0 1ρ = , the corresponding value of 1P being 1 7 746P = − . .

22 2 a

M De e e

M D ek k kq q q

φ

φ

∞

= , = , =

Physical Quantities

BPS condition

22 2 2 2

2

2(1 ) e

aa M D Qa

+ = .+

0a → and cra a→ we find out, that

21 2(1 )1

e

aM a MD Q

α − /+→ , →

Decoupling of dilaton (a=0)

0 0a α→ , →

We redefine free parameter of expansion on horizon like:2

0 11

aP ρ ρα= −

Also we have:4024 eq

ρα = and 20

4e Haφ

ρα =

22 3 41 1

2 3 40 0 0

0 1

1 2 3( ) ( ) ( ) (5)

( )

H H H

H

w r r r r r r O

r r

ρ ρρ ρ ρ

ρ ρ ρ

= − − − + − +

= + −

In the limit:expansion on horizon transform to:

what is expansion for extremal (for some ) RN solution with charge

1ρ2

0 e Haeq φρ=

Decoupling of GB term (critical value of a)

When dilaton coupling is tending to critical value, we have growing mass off solution.

Increasing of mass of solution from other hand may be interpreted(from definition of mass), as decreasing of gravitational constant Gfor the considered system.

But Gauss-Bonnet term doesn’t affected of this changing, so his role in action decreasing and finally neglected in limit of infinite value of mass.

Thermodynamics property of solution

20

1 1 ( ) 02 2 H

H

ttrrH r r

r r

gT g r r

rπ πρ =

=

⎛ ⎞∂= = − | =⎜ ⎟⎜ ⎟∂⎝ ⎠

Temperature:

Entropy:

• classical formula (without accounting of GB term)

• Sen’s Entropy function (hep-th/0506177)simplified Wald’s formula for AdSxS

20S πρ=

202S πρ=

• It is exist electrical charged extremal BH solution

with a non-zero radius of horizon for D=4 a DGB gravity.

•Entropy for this solution is twice as defined by Berkenstein

formula.

• Last result consists with results for a supergravity theory.

Intervals of critical dilaton coupling constant (from expansion on horizon) is divided by root of equation from denominators of horizon expansions:

In the limit of big magnetic charge electrical charge tends to zero, also dilaton field vanishe and solution become RN solution(with finite radius of horizon).

2 20

2

42(2 )

me

m

qq

qα ρα+

= .+

•Dyonic solution

2 4 2 2 2 2( ) (6 4 ) 4 ( ( 2) ) 0,j m j m j ja a q a a q a a aα α− + − + + − =

Results of numeric integration

Regions of asymptotically flat solution (black colour) and solution with linear dilaton background behaviour (red colour)for small value magnetical charge and dilaton coupling constant. For LDB case an electric charge belong to background.

LDBsector

•Entropy: 22 2 0

0 222 42e m

m

S q qq

απρπ α πρα

= + = ++

•The global parameters of solution (mass, dilaton charge, asymptotic values of dilaton ) define from electrical charge like:

where ratios depend only from dilaton constant and magnetic charge.

22 2

( , ) ,a

M m De e e

M D ek a q k kq q q

φ

φ

∞

= , = , =

but result arises another question about a dependence entropy of surface of black hole.

•Near critical points we have the similar BPS condition for the global parameters of solution:

22 2 2 2 2

2

2 ( )(1 ) e m

aa M D Q Qa

+ = ++

( )cra a→

Critical value of solution can has connection with transition between black holes and strings (Cornalba, Costa, Penedones and Vieira, “From fundamental strings to small black holes,”

JHEP (2006) 023; hep-th/0607083)

Classical limits:

•dilaton coupling constant tends to zero , in this case GB term decouple and solution become known dyonic RN solution,

•magnetic charge tends to infinity and the role of GB is neglected, and we get RN-solution with infinity magnetic charge for every coupling constant of dilaton except , in the last cases solution is dyonic dilatonic black hole.

0a →

( 0)mq α→∞ →

2 ( 1)2

n na +=

1. Even first higher order curvature term introduce radically new properties of solution, what allow resolve a puzzle with zero radius and entropy of extremal black hole in the case of a BH solution with one charge and extend range of dilatonic coupling for solution in dyonic case.

2. The critical state of system which satisfy BPS equation on global parameter may have relation to transition betweenblack holes and strings, and have to further invistigation.

Summary and recent results

• The model was investigated in the Einstein frame, also possible to consider a coupling of GB term in a string frame.

• Including GB term with a exponential of dilaton coupling breaksS-duality of system and entropy doesn’t just depend on surface area of horizon (dyonic case). Possible solution is considering more complicated case, in which coupling of GB term is based on a Dedikind function, which preserves this symetry.

Gal’tsov, Davydov, 0812.5103

• For investigated D=4 system AF solution doesn’t exit for a dilaton coupling a=1, what inconsist with known from a string theory. Our recent investigation shows that for D>4a range of dilaton coupling is extended and also revives some interesting properties, which expect futher consideration.